Problem: All of the 5th grade teachers and students from Santa Rita went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$2.00$ each for students, and the group paid $$48.00$ in total. The next month, the same group visited a science museum where the tickets cost $$17.00$ each for teachers and $$10.50$ each for students, and the group paid $$141.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+2y = 48}$ ${17x+10.5y = 141.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-17x-4y = -96}$ ${17x+10.5y = 141.5}$ Add the top and bottom equations together. $ 6.5y = 45.5 $ $ y = \dfrac{45.5}{6.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {8.5x+2y = 48}$ to find $x$ ${8.5x + 2}{(7)}{= 48}$ $8.5x+14 = 48$ $8.5x = 34$ $x = \dfrac{34}{8.5}$ ${x = 4}$ You can also plug ${y = 7}$ into $ {17x+10.5y = 141.5}$ and get the same answer for $x$ ${17x + 10.5}{(7)}{= 141.5}$ ${x = 4}$ There were $4$ teachers and $7$ students on the field trips.